Surface wave luneberg lens antenna system



CROSS REFERENCE Oct. 22, 1963 3,108,278

C. H. WALTER SURFACE WAVE LUNEBERG LENS ANTENNA SYSTEM Filed Dec. 1, 1958 3 Sheets-Sheet 1 ELOCITY RATIO 0.! 0.2 0.3 THCKNESS OF DELECTR'C SHEET oowmm FIRST ORDER gd ilj lxz'w n- MODE K @wm -w INVENTOR CARLTON H. WAIJ'ER MODE SURFACE IAVE LUNEBERG LENS ANTENNA SYSTEM Filed Dec. 1, 1958 3 Sheets-Sheet 2 azcoaozn F 5 v AMPLIFIER I v FE: B a [N1 ENTOR.

BY cmu'ou H.WALTER Oct. 22, 1963 c. H. WALTER 3,108,278

SURFACE IAVE LUNEBERG mus ANTENNA sysma Filed Dec. 1, 1958 3 Sheets-Sheet 3 III-rm]. "I

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CARLTON H. WALTER United States Patent 3,108,278 SURFACE WAVE LUNEBERG LENS ANTENNA SYSTEM Carlton H. Walter, Columbus, Ohio, assignor to The Ohio State University Research Foundation, Columbus, Ohio, a corporation of Ohio Filed Dec. 1, 1958, Ser. No. 777,524 11 Claims. (Cl. 343-753) My invention relates broadly to radar antennas and more particularly to a surface-wave structure of dielectric material which performs as a Luneberg lens.

Basic theoretical work by R. K. Luneberg on the optics in a medium of variable index of refraction as set forth in his publication The Mathematical Theory of Optics, Brown University Press, Providence, R.I., 1944, resulted in a type of lens that has many applications in microwave antennas. Luneberg showed that if a dielectric sphere of unit radius has an index of refraction 1 satisfying the relation where p is the distance from the center of the sphere, then a plane wave incident on the sphere would focus at a point on the surface of the sphere diametrically opposite from the incident plane wave.

A great deal of work has been done recently on microwave structures of both spherical and cylindrical shapes having the radial variation in 7] given by Equation (Eq.) 1. Emphasis has been placed on techniques for obtaining the necessary radial variation in 1 as well as on modifications and applications of the lens. The electromagnetic theory of the Luneberg lens has been worked out by H. Jasik for the cylindrical lens in his work entitled, The Electromagnetic Theory of the Luneberg Lens, Report TR 54-121, Air Force Cambridge Research Center, Bedford, Mass, November 1954, and by C. H. Wilcox and C. T. Tai for the spherical lens in their respective publications, The Refraction of Plane Electromagnetic Waves by a Luneberg Lens, Report MSD 1802, Lockheed Aircraft Corporation, June 1956, and The Electromagnetic Theory of the Spherical Luneberg Lens, Report 667-17, The Ohio State University Research Foundation, August 1956. J asik solved for the far field of the cylindrical lens and obtained numerical results for both omnidirectional and dipole sources. Jasik found good agreement between the results of his exact solution and the results he obtained by optical methods for a lens diameter on the order of 3 wave-lengths. Wilcox solved for the fields at or near the focus for a plane wave incident on a spherical lens. Tais solution for the spherical lens is more general; it can be used to find the far field with excitation at the focus or the field near the focus for a plane wave incident on the lens. Recent work by E. H. Braun disclosed in the publication Radiation Characteristics of the Spherical Luneberg Lens, IRE Transactions on Antennas and Propagation, volume AP-4, No. 2, April 1956, on the spherical Luneberg lens gives the beam width, gain and side lobe level of the far-field pattern for various distributions of electric and magnetic fields over the surface of the lens.

Another basic study that has been applied to microwave antennas recently is that of surface-wave propagation. An electromagnetic surface-wave can be defined as an electromagnetic wave that propagates along an interface between two media. The earliest work on this subject appears to be that of A. Sommerfeld, Fortpfianzung Electrodynamischer Wellen an einem zykindrischen Leiter, Ann. Phys. U. Chemil, vol. 67, who discussed the propagation of a transverse magnetic surface wave along an infinitely long cylindrical wire of finite conductivity.

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Important contributions have been made by C. C. Cutler with his work Electromagnetic Waves Guided by Corrugated Conducting Surfaces, Report MM-44-160-2l8, Bell Telephone Laboratories, 1944; G. Goubau with his work Surface Waves and Their Applications to Transmission Lines, Journal of Applied Physics, volume 21, 1950, on electromagnetic waves guided by a dielectric coated wire, and S. S. Atwood with his work on Surface- Wave Propagation Over a Coated Plane Conductor, Journal of Applied Physics, volume 22, 1951. A good summary and an extensive bibliography on surface waves have been presented by F. J. Zucker in his paper The Guiding and Radiation of Surface Waves, Proceedings of the Symposium on Modern Advances in Microwave Techniques, Polytechnic Institute of Brooklyn, New York, November 1954.

It is the purpose of the present disclosure to show that a surface-wave structure can be made to perform as a Luneberg lens. In particular, it will be shown that a circular dielectric sheet on a ground plane can be made to perform as a Luneberg lens in the plane of the sheet and at the same time perform as an endfire antenna in the orthogonal plane. An approximate analysis will be presented that enables one to compute the radiation pattern of this surface-wave Luneberg lens antenna.

One of the objects of my invention is to provide a radar antenna which can be mounted flush with the outside skin of aircraft or missiles.

Another object of my invention is to provide a radar antenna of circular dielectric sheet construction on a ground plane which performs as a Luneberg lens in the plane of the sheet and at the same time performs as an endfire antenna in the orthogonal plane.

Another object of my invention is to provide a surfacewave Luneberg lens radar antenna capable of 360 degrees scan either by rotating the feed around the rim of the lens antenna or by fixing the feed to the lens and mechanically spinning the lens antenna.

Still another object of my invention is to provide a radar antenna which permits 360 degrees of substantially instantaneous scan and requires no cumbersome mechanical scanning mechanism.

A further object of my invention is to provide a flush mounted radar antenna which can be converted into a flush mounted radar reflector for use in radar echo enhancing applications.

Still a further object of my invention is to provide a radar antenna for large ground based radar systems with less frontal area, and less moving mass in scanning operation.

Other and further objects of my invention reside in the method of obtaining the necessary variation in the index of refraction for the Luneberg lens through proper control of the velocity of the surface-wave as set forth more fully in the specification hereinafter following by reference to the accompanying drawings in which:

FIG. 1 is a schematic drawing showing the coordinate system for an infinite dielectric sheet of thickness d and dielectric constant 6 on an infinite, perfectly conducting ground plane;

FIG. 2 is a graphical presentation showing a graphical solution of Equations 19 and 23 and showing occurrence of higher order modes;

FIG. 3 is a graphical illustration of velocity ratio versus thickness of dielectric sheet for a polystyrene sheet on an infinite, perfectly conducting ground plane for the dominant transverse magnetic mode;

FIG. 4 is a graphical illustration of thickness of dielectric sheet on ground plane versus normalized radius for a transverse magnetic surface-wave Luneberg lens;

FIG. 5 is a fragmentary enlarged view partly in cutaway section of a tapered-depth antenna used as a feed element for a transverse magnetic surface-wave lens;

FIG. 6 is a top plan view of a surface-wave Luneberg lens antenna of my invention showing the lens and feed element on a finite ground plane;

FIG. 7 is an enlarged side elevational view of the surface-wave Luneberg lens antenna shown in FIG. 6, but with a portion of the ground plane omitted;

FIG. 7a is an enlarged cross sectional view taken substantially along line 7a7a of FIG. 6;

FIG. 8 is a schematic diagram showing the arrangement in block form for measuring antenna patterns of the surface-wave Luneberg lens antenna;

FIG. 9 is a schematic diagram of the coordinate system for the surface-wave Luneberg lens antenna;

FIG. l0 is a schematic diagram of the coordinate system used in the approximate analysis of the surface-wave Luneberg lens antenna;

FIG. 11 is an illustration in schematic form of the geometry for finding phase along b in terms of ray direction;

FIG. 12 is a graphical presentation illustrating the phase delay along b for the 18 inch diameter Luneberg lens at a wavelength A =3.10 cm.;

FIG. 13 is a graphical presentation of the computed far-field patterns of tapered-depth antennas used as feed elements on the transverse magnetic surface-wave Luneberg lens antenna;

FIG. 14 is a graphical illustration of the amplitude distributions along b for the 18 inch diameter Luneberg lens at a wavelength 7\ ,=3.l0 cm.;

FIG. 15 is a graphical plot of rays through a unit radius Luneberg lens with the source at p =0.85l;

FIG. 16 is a schematic illustration of the geometry for computing the far-field pattern of a line source of current;

FIG. 17 is a graphical presentation showing computed far-field patterns comparing a continuous source with several arrays for L= A(l)=A =l, and (l)= =0;

FIG. 18 is a top plan view of the recessed lens modified form of my invention;

FIG. 19 is a side elevational view of the recessed lens form of my invention particularly showing the manner in which the ground plane is dished to receive the lens antenna;

FIG. 20 is a longitudinal sectional view taken substantially along line 3030 of FIG. 18;

FIG. 21 is a perspective view of the end portion of the waveguide feed element for the recessed lens antenna form of my invention and particularly showing the manner in which the terminating end of the waveguide is curved to fit flush with the convex surface of the recessed lens;

FIG. 22 is a top plan view of the portion of waveguide feed element shown in FIG. 21;

FIG. 23 is a top plan view of a modified surface-wave Luneberg lens antenna of my invention showing the feed element positioned well in from the rim of the lens; and

FIG. 24 is a side elevational view of the lens and feed configuration shown in FIG. 23.

My invention is directed to a surface-wave Luneberg lens antenna which is inherently a flush-mounted antenna; that is, it can be made perfectly flush with respect to the outside surface of the structure in which it is mounted and nearly flush with the interior surface. The antenna is capable of 360 degrees scan either by rotating the feed around the rim of the lens or by fixing the feed to the lens and spinning the lens. Because of the geometry of the antenna, extremely rapid scan rates are possible.

An antenna of this type can be built into an air frame or designed as a thin package to be fastened to the surface of an aircraft. It can be used in most conventional radar systems such as radar navigation, ground mapping radar, collision warning radar, and early warning radar. Although originally conceived for aircraft use. it would offer certain advantages over conventional reflector antennas in ground based radar. Two main advantages which become very important for large antennas are less frontal area and less moving mass in scanning.

The surface-wave Luneberg lens antenna of my invention is a lens with focus at the rim. Thus a radar picture is available at the rim of the lens and an array of feed elements around the rim may be used to obtain this picture. This permits the antenna to be used in instantaneous direction-finding systems and in instantaneous-scan radar systems. If the array of feed elements is short circuited, the antenna becomes a good, flush-mounted reflector useful in many radar echo enhancing applications. An example would be that of increasing the nose-on radar return from an aircraft to aid ground controlled landing systems.

ANALYSIS OF A DIELECTRIC SHEET ON A GROUND PLANE By definition the index of refraction n is given by where c is the velocity of light in free space and v is the phase velocity of a wave in the medium under consideration. From Equations (Eqs.) 1 and 2 we have The design of a surface-wave Luneberg lens will be based on Eq. 3 where v will be taken as the phase velocity of the surface-wave.

The surface-wave structure to be considered is that of a dielectric sheet on a ground plane. It is necessary to analyze this structure to obtain v as a function of the thickness and dielectric constant of the sheet for a given mode of propagation. For the purpose of an approximate pattern computation later, it will be necessary also to find the tangential electric and magnetic fields (i and 11, respectively) at the dielectric-air interface.

In order to obtain Relation 3 for a given mode, it will be necessary to vary the thickness and/or the dielectric constant of the dielectric sheet. It will be assumed that the variations are sufficiently gradual so that at any point v, E, and 1 1} are essentially the same as they would be for a sheet of that thickness and dielectric constant and of infinite extent. This reduces the analysis to that of an infinite sheet of dielectric on a ground plane of infinite extent.

The infinite sheet problem has been considered by Attwood and Zucker in their previously mentioned works. Zucker applied a transverse resonance method to obtain propagation constants for both transverse magnetic (TM) and transverse electric (TE) modes. This method does not give the field configurations, however. To find these it is necessary to apply Maxwells equations and the appropriate boundary conditions.

The results of Zucker and Attwood show that surfacewaves can propagate in TM and TE modes. Furthermore, each TM mode has only one rectangular component of E and each TE mode has only one rectangular component of E. An analysis is given here that is based on a suitable assumption for the single field component for each of the two cases. This quickly leads to expressions for the field configurations and the propagation constants.

TM Surjace-Wave Mode A solution for TM surface-wave propagation in the x direction for the structure shown in FIG. 1 can be obtained by assuming ii to be of the form in medium 1 which represents an infinite dielectric sheet of thickness d and dielectric constant e" on an infinite, perfectly conducting ground plane 2, and

H y component of the vector fi =unit vector in y direction A and A =complex constants k =phase constant in the z direction in medium 1 k =attenuation constant in the z direction in medium k =phase constant in the x direction.

The electric and magnetic fields that exist on the structure in FIG. 1 must satisfy Maxwells equations and the appropriate boundary conditions. The sheet 1 in FIG. 1 is assumed to be lossless, isotropic and homogeneous. Assuming a source free region and time variation of the form e we have for Maxwells equations where n is permeability and w is angular frequency.

Expressions for E in media 1 and 0 are obtained by means of Eq. 7. Performing the curl operation results in Equation 11 shows that Eq. 14 is satisfied. Equation 15 is satisfied when A A (0)k (1) (17) tnk m Equation 16 is satisfied when (18) A =A cos [k d] Combining Eqs. 17 and 18 results in sin [k d] Equation 20 is obtained by substituting Eq. 7 into Eq. 6 and making use of Eq. 9. Substituting Eq. 4 into Eq. 20 gives where 6 Substituting Eq. 5 into Eq. 20 gives Combining Eqs. 21 and 22 and assuming that #()=/L(1) gives Equations 19 and 23 can be solved for 1: or k Knowing either k or k enables us to find k by means of Eq. 22 or Eq. 21, respectively.

The periodicity in Eq. 19 implies that there is more than one mode of operation. This is illustrated by the graphical solution of Eqs. 19 and 23 shown in FIG. 2. Only waves which decay exponentially in the z direction in the air medium are of interest; hence, only positive values of k need be considered. No loss in generality occurs in considering only positive values of k The intersection of the circle and tangent curve in FIG. 2 give the value of k More modes become possible as the radius of the circle increases. It is interesting to note that the dominant TM mode has no cut-oft". As d/h decreases to zero the TM mode degenerates into a TEM (transverse electromagnetic) wave propagating along the ground plane. In the design of a surface-wave Luneberg lens only the dominant TM mode will be considered, although higher mode lenses are possible.

For a lens design involving Eq. 3 it is necessary to work with the velocity ratio c/v. It has been assumed that the fields in the structure of FIG. 1 travel unattenuated in the x direction with phase constant k which may be expressed as where X, is the wavelength in the x direction.

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Further- Equation 21 now becomes From Eq. 22 we obtain an alternate expression ft 1+ in) It is possible to solve for c/ v directly by solving Eqs. 27b

and 28 for k and k respectively, and substituting into Eq. 19.

In addition to knowing c/ v as a function of d and e it is necessary to know tangential Ti and E at the airdielectnc interface in order to compute far-field radiation patterns. These components are obtained from Eqs. 11 and 14 evaluated at z=d.

TE Surface-Wave Mode A solution for TE surface-wave propagation in the x direction for the structure in FIG. 1 can be obtained in a manner similar to that for the TM case above. The solution is obtained by assuming E to be of the form in medium 0, where B and Bill) are complex constants. The following components of H in media 1 and are obtained by means of Eq. 6:

Assuming that a the boundary conditions at the interface require that Combining Eqs. 37 and 38 gives (39) k =k cot [k d] Equations 21, 22, 23, 27b, and 28 also apply to the TE modes thus the velocity ratio c/v can be obtained from Eq. 39 for the TE case in the same manner as from Eq. 19 for the TM case. Tangentia1 E and E at the airdielectric interface are obtained from Eqs. 29 and 32 evaluated at z=d.

A study of Eq. 39 shows that more than one mode is possible. However, in the TE case the lowest order mode does have a cut-off whereas in the TM case the lowest order mode does not. For a given dielectric there is a critical value of d/k below which TE surface-wave propagation is not possible for the structure in FIG. 1. This does not mean that the structure of FIG. 1 is not suitable for a TE surface-wave Luneberg lens; it merely means that such a lens would have finite thickness at the rim.

EXPERIMENTAL RESULTS In order to test the validity of the surface-wave lens concept, I designed, constructed and tested a surfacewave Luneberg lens. The design was based on the dominant TM surface-wave mode previously set forth.

Design of Surface-Wave Luneberg Lens The necessary radial variation in c/ v for the Luneberg lens is given by Eq. 3 in terms of the normalized radius p. This variation in c/ v was obtained by varying d with 6 held constant. For the TM surface-wave the ratio 0/ v as a function of d is found from Eqs. 19, 27, and 28. This results in an equation of the form 8 Substituting Eq. 3 into Eq. 4011 gives (400) (1) d W tan A graph of c/v versus d/A from Eq. 40b is shown in FIG. 3 for polystyrene dielectric (e /e =2.5. A graph of d/x versus p from Eq. 40c is shown in FIG. 4 for four difierent values of e /e From the graph in FIG. 4 for experimental purposes and in order to prove the correctness of my surface-wave lens concept a polystyrene lens 18 inches in diameter was shaped on a lathe. The free space wavelength A was chosen to be 3.10 centimeters. I desire that it be understood that the above figures are not meant to limit the scope of my invention but are used merely by way of explanation.

Method of Feed In the present case I have specified a TM surface-wave. It is necessary to find a feed, or exciting structure, that will produce this surface wave. Although there ar some applications such as pattern shaping where direct radiation from the feed is quite useful, the ideal feed for this application would have the same field configuration and propagation constant as those of the surface-wave structure where the feed is located; and, in addition, the feed would be perfectly coupled to the surface-wave structure in the sense that all of the power put into the feed is transferred to the surface-wave.

Although an ideal feed for a surface-wave structure would have the same phase velocity along its length as the surface-wave structure where the feed is located, this may be undesirable in a lens because of defocussing. Ideally a lens should be excited at the point of focus. Thus a certain amount of compromising may be necessary in a surface-wave lens since practical feeds for surfacewave structures are generally several wavelengths in extent.

A structure that appears to be a very good compromise as a feed element for a surface-wave Luneberg lens is the tapered-depth endfire antenna. This type antenna is described in the publication Endfire Slot Antennas, IRE Transactions on Antennas and Propagation, vol. AP-3, No. 2, April 1955, by B. T. Stephenson and C. H. Walter. I have shown a sketch of such an antenna 6 suitable for exciting the TM surface-wave Luneberg lens in FIG. 5. The aperture of the antenna in FIG. 5 designated at 3 has the necessary field components (H E and E for TM surface-wave excitation and the velocity ratio c/v along the aperture may be controlled by varying the depth. The antenna 6 of FIG. 5 may be placed under the lens 4 at the rim as shown at 5 in FIGS. 6, 7 and 7a and the phase velocity adjusted to match that of the lens near the rim. The antenna 6 consists of a waveguide member 7 tapered at one end thereof as shown in FIGS. 5 and 70, said tapered end being filled with a dielectric material as shown at 8.

M easurem ents Pattern measurements were made at a wavelength of 3.10 cm. on the polystyrene lens previously described. Two separate tapered-depth feed antennas were used, one with aperture 2 wavelengths long and the other with aperture 3 wavelengths long. Each feed consisted of polystyrene-filled X-band waveguide with depth uniformly tapered to zero over the length of the aperture section as shown in FIG. 5. No claim is made that the linear taper gives the optimum feed element, however these feed elements perform quite well and the patterns of the feedlens combination can be predicted as will be later described.

The feed element was mounted in a sheet of Az-inch thick aluminum which served as a ground plane 9. The polystyrene lens 4 was taped to the ground plane 9 with the feed at the rim of the lens as shown at 5 in FIGS. 7 and 7a. The ground plane was mounted on a rotating tower and the surface-wave Luneberg lens was used as a receiving antenna in the pattern measuring arrangement shown by the block diagram in FIG. 8. This is a conventional pattern measuring system with the exception of the selective amplifier 10 and the wide range square root recorder 11 which are special units that have been designed at The Ohio State University Antenna Laboratory. The recorder 11 is capable of plotting very accurately the square root of its input signal for an input range of 80 db. The square root function is necessary to compensate for the characteristic of the crystal or bolometer detector that is used in this system. The resulting pattern plot from the recorder is proportional to the electric field.

The coordinate system that I will use in describing the measurements is shown in FIG. 9. In general the planes 6=90 (x-y plane) and I =0 (x-z plane) are of principal interest.

It is a characteristic of an endfire antenna in a ground plane that the pattern in the plane of the ground plane essentially is not affected by the extent of the ground plane whereas the pattern in the orthogonal plane =0) is very dependent on the extent of the ground plane. This, of course, applies to the surface-wave Luneberg lens since it inherently is an endfire device.

It is quite evident from measurements taken that the surface-wave structure is providing lens action in the plane of the lens and at the same time performing as an endfire antenna in the orthogonal plane.

If a conventional two-dimensional (9:90" plane) Luneberg lens of the diameter in wavelengths that we have been considering were fed at the rim with a small horn having a cos i field pattern, the resulting pattern in the 9:90" plane would have a half-power beam-width of approximately 4 degrees and a side lobe level on the order of 19 db below the beam maximum. The surfacewave Luneberg lens with 2 wavelength tapered-depth feed has a half-power beam-width of about 5 /2" with side lobes on the order of 20 db below the beam maximum.

APPROXIMATE ANALYSIS OF ANTENNA PATTERNS By means of certain assumptions, an exact solution for an infinite sheet of dielectric on a perfectly conducting ground plane of infinite extent has been used in designing a surface-wave Luneberg lens. The radiation characteristics of this lens with tapered-depth feed elements have been measured. The measurements indicate that the feed-lens combination works satisfactorily. A lens pattern is obtained in the plane of the lens, and in the orthogonal plane the pattern is that of an endfire traveling-wave antenna with infinite ground plane. Patterns in both of these planes are affected by the length of the feed element. In particular the pattern in the plane of the lens indicates a substantial increase in defocussing as the length of the feed increases. It appears to be worthwhile to attempt an analysis of the surfacewave Luneberg lens with tapered-depth feed in order to present a better understanding of how the feed affects the radiation patterns.

An exact analysis is beyond the scope of this application; however, such an analysis might be attempted by extending Jasiks transmission line mode treatment of the cylindrical lens, as set forth in his work previously mentioned, to the surface-wave modes in th present case.

In the approximate analysis to be presented, two problems are considered. The first concerns the computation of the far-field radiation pattern of the lens-feed structure in the plane of the lens. The second concerns the computation of the far-field pattern in the orthogonal plane through the lens-feed axis. These are the two principal planes in which measurements were made as previously set forth and will again be designated as 9=90 and i =0 plane, respectively.

Pattern in G= Plane The method used in computing the pattern in the 9=90 plane (plane of the lens) is essentially an extension of a method used by G. D. M. Peeler and D. H. Archer in their report A Two-Dimensional Microwave Luneberg Lens, Report 4115, Naval Research Laboratory, Washington, DC, March 1953, and considered briefiy by Jasik in his previously cited report.

A two-dimensional (x-y plane) lens is assumed where, by means of optical methods, an electric field distribution in both amplitude and phase is found along the line b (a segment of the line x=l) in FIG. 10. The distribution along the line b can be thought of as an equivalent broadside source replacing the lens and feed as far as the pattern in the plane of the lens is concerned. The radiation pattern is simply that of a broadside radiator with the appropriate amplitude and phase distributions.

Point Source at Rim of Lens When a point source is located at the rim of the lens the problem is relatively simple. The electrical path lengths from the source to points on b are equal (all elements of the broadside radiator are in phase) and all rays emerge parallel to the x axis. The amplitude of the field along b is found by considering energy flow based on geometrical optics. For the two-dimensional lens (no z variation) all of the energy in an incremental element of angle A5 at the feed passes through an incremental segment Ay along b (see FIG. 10). It is therefore necessary that where S(B)=cnergy distribution per unit angle in the pattern of the feed. S (y)=energy distribution per unit length along b.

We have assumed that there is no z variation, and with the feed at the rim all rays are parallel to the x axis therefore b(y)= y(0) where S,(%) is the energy density along y at the rim of When the incremental elements A and Ay are replaced by the differential elements dB and dy, respectively, Eq. 41 gives The field strength is proportional to the square root of the energy density, therefore 47 cos 6 where E (y)=electric field distribution along b. E(fi)=electric field pattern of the feed.

Although the source has been assumed to be a point source, Eq. 47 has proven to be a useful approximation 11 for an extended source, such as a horn, whose aperture is tangent to the rim of the lens.

Point Source Within the Lens For a surface-wave lens such as that used earlier in this disclosure for explanation purposes it is necessary to consider a source extending into the lens. In this case defocussing could be expected and measured results disclosed earlier in this application indicate that defocussing does occur.

Since it is possible to consider a born at the rim of a conventional Luneberg lens as a point source, it will be assumed that the extended feed in the present case is effectively a point source with the same energy distribution as the actual source. However, difiiculties arise in determining the position of this effective point source since it will lie at some point I within the lens (see FIG. 10).

The point at which the feed is assumed to exist may be found by considering the lens and feed as coupled waveguides with non-uniform phase velocities. It is assumed that the individual phase velocities are unaltered by mutual coupling. Maximum transfer of energy from the feed to the lens is assumed to take place where the velocities are equal.

Applying this method of locating an effective point source to the two sources considered in the section of this application entitled Experimental Results, we find from Eq. 3 and from reference data on endfire slot antennas that the 3 wavelength source is effectively at p =0.851 and the 2 wavelength source is effectively at p =0.894.

For a source at P in FIG. the phase distribution along x=l can be found either by means of the path length along the ray from P to Q (see FIG. 10) or by means of the angles that the rays make with the line b.

Considering first the path length in a medium of index of refraction 1 the optical path length p is given by where a's is a differential element of path length along the ray through P and P By Fermat's principle the path or ray from P to P is such that p is a minimum. It can be shown by the calculus of variations that an extremal (minimum value) of the integral in Eq. 48 (whose integrand is without explicit dependence on 4:)

where -r is the angle the ray makes with a radial line at the rim (see FIG. 10). An alternate expression for Eq. 49 is where 7 is the angle between a radial line and a tangent to the ray (see FIG. 10).

Choosing P as P the point where the source is located (see FIG. 10), we have from Eq. 50

17p sin =sin r 'n p sin fl=sin 1- where =index of refraction at point P =radial distance for P fi=angle at which ray emerges from source.

sin 6F Equation 53 may 5 p=f"'- P1 p p-( 1tpt sin 1 f P p p p (mp5 sin 6V After substitution of variables, the integrals in Eq. 54 are readily found in standard tables and the resulting expres sion for p is given by P.. 17 17 p, COS Bs1n 2 1 -Pt 1 Pt (v pt i where for p decreasing along the path the third term is negative and the fourth term is equal to or less than 1r/2 and for p increasing the third term is positive and the fourth term is equal to or greater than rr/Z.

The total path from P to Q is given by In order to evaluate Eqs. 56 and 57 it is necessary to know and 1- for a given ray Starting with Eq. 40 we obtain 1Y (sinr which can be expressed as sin 'rdp condition that =1r at p=p The resulting expression for is given by Thus for a ray starting at angle ,3 at p we find from Eq. 51 and hence from Eq. 61. The path length in degrees from P to Q is given by where pA is the radius of the lens in free-space wavelengths.

An alternate expression for the phase distribution along b can be obtained in terms of the angle that a ray makes with the line b. From FIG. 11 which shows rays emerging from a segment of b we have the relations k =free space wavelength. x =wavelength along b.

13 In terms of the phase velocity v along b we have where =phase in degrees. y =reference point.

Equation 65 can be evaluated graphically when necessary and with respect to a given reference point on b gives the same results as 62 within the accuracy of the graphical solution.

The amplitude of the field along b for a source at p 1 is found from Eq. 42 as in the perfectly focussed case (p =l); however, it is necessary to find dy/dfl for Ps From Eq. 57 we have y as a function of and 1'. From Eq. 51 we have 1- as a function of 5 for a fixed p From Eq. 51 we have as a function of ,B for a fixed p The differential element dy can be written as Substituting into Eq. 66 and replacing 'n p sin ,9 by sin r we have COS 1' where the bracketed quantity is obtained from Eq. 71. For p =1 and fi= Eq. 72 reduces to Eq. 44.

Expressions have now been obtained that enable us to find an amplitude and phase distribution along b which is assumed to replace the lens and feed as far as the pattern in the plane of the lens is concerned. The two cases for which measurements have been obtained will be considered. Patterns will be computed by an approximate method described later in this application. The phase delay ,0 as a function of y along b has been computed by means of Eq. 65 for p =0.851 (3 wavelengths feed) and p =0.894 (2 wavelengths feed). The

hm Thus for p l 0 lens could be recessed for the TM 14 results are shown in FIG. 12 (the distributions are symmetrical about y=0).

The amplitudes of the fields along b for these two cases are found from Eqs. 42 and 72. For E( 8) in Eq. 42 we use the magnitude of the far-field radiation pattern of the feed element by itself. The far-field patterns of the two sources are shown in FlG. 13. Data for computing these patterns were taken from the work by Stephenson and Walter entitled Endfire Slot Antenna previously cited in this application. The resulting amplitude distributions along b (the distributions are symmetrical about y=0) are shown in FIG. 14.

An interesting problem arising from the present study is that of determining the appropriate index of refraction in a modified Luneberg lens to optimize the focussing of a given extended-source feed such as the tapered-depth feed considered above.

Pattern in r =0 Plane The determination of the pattern in the I =O plane (see FIG. 9) is more difficult than that for the 6=90 plane. The use of an equivalent line source appears to be impractical. Instead, it is necessary to consider radiation from a large part of the lensground plane surface.

Before going into an approximate pattern computation in detail, it is well to set up an exact expression for the far-field pattern in terms of surface currents. By Huygens principle the electromagnetic waves generated in a source free region by any set of sources can be generated by a sheet of electric currents of surface density j ixlf and a sheet of magnetic currents of surface density 17=F H spread over the surface S which surrounds the sources, where n is a unit vector normal to S and pointing into the source free region. Two vector potentials may be defined in the usual manner 73 Z- f S T dS and h i 37 -51 ((4) 41rJ] S 1' (is where k=propagation constant.

n=permeability.

e=dielectric constant.

r=distance from a point on S to point of observation.

The electric field intensity on the source free side of S is given by where w is the angular frequency of the source.

For the surface-wave Luneberg lens on a finite ground plane S will be taken as the surface of the feed-lensground plane structure. If the actual tangential E and E on S were known, then Eq. 75 would give the exact field at any point outside S and in particular the field in the 9:90 and I =O planes. The exact evaluation of tangential E and F over S and the exact solution of Eq. 75 would be prohibitively difficult.

The problem can be greatly simplified by assuming that the currents on the rear surface of the antenna are negligible compared to those on the front surface where the lens is located. It is further amumed that the front surface is a plane surface. This should be a reasonable assumption since the lens protrudes but a short distance above the ground plane (see FIG. 4). In fact in practice the case to provide a perfectly fiat surface. Tangential E and E over the simplified surface may be found approximately by optical methods.

Consider again energy fiow along optical paths through the lens as in FIG. 10. It will be assumed that the fields in the lens at a cross section of width AW are the same as if it were a section of an infinite sheet of dielectric of the same thickness as within AW and with the dominant TM wave propagating in the direction of the rays. It will be further assumed that the energy density is proportional to (|A where |A is the amplitude of the magnetic field in the dielectric region for the TM case. Thus we can write where W may be found graphically as described later. The relative magnitudes of tangential E and H (E; and E respectively) can be obtained from Eqs. 11 and 4. The magnitude of E from Eq. 11 gives the magnitude of E. The magnitude of 111 from Eq. 4 gives the magnitude of fi As indicated in FIG. 10, E is parallel to the ray and I? is normal to the ray.

For the far-field pattern in the I plane, contributions from the y component of E cancel and contributions from the x component of fi cancel due to symmetry (see FIG. 9 for coordinate system). Thus the far field in the I :0 plane can be expressed in terms of magnetic current density lil -=15 and electric current density J =H over the surface of the lens-ground plane. Since the ground plane has been assumed to have perfect conductivity, E will be zero everywhere except on the lens surface. fi may exist over the entire lens-ground surface.

In addition to the amplitudes of the surface currents, it is necessary to know the phase at any point on the surface with respect to some reference point. With respect to p the phase at any point on the surface of the lens is given by Eq. 55. In order to evaluate Eq. 55 at some point in the lens it is necessary to know atthat point and the angle ,6 which determines the ray going through that point. The coordinates p and B for a given point may be found from a plot of the ray paths. This is readily done by means of Eqs. 60 and 51. A ray path is obtained by plotting versus p with held constant. The resulting graph for a source at p 0.85l and 13 in 5 increments is given in FIG. 15. This graph in addition to giving p and 5 for use in Eq. 55 for path length, also gives A S/AW for use in Eq. 76.

Thus by optics the amplitude and phase of i and fi may be found atany point on the surface of the lens. The amplitude and phase of i on the ground plane in front of the lens may be found by extending the rays as straight lines after they emerge from the lens.

Even though we have reduced S to a finite plane surface it still would be extremely difficult to obtain explicit relations for fi and T and perform the integrations called for in Eqs. 73 and 74. The problem is greatly simplified if the surface distribution is approximated by a planar array of electric and magnetic dipole elements having the same amplitudes and phases as the surface currents at the points where the dipoles are located. Based on the approximate pattern computations set forth later in this application, it appears that a reasonably good approximation would be obtained in the region of the main beam for as much as a wavelength separation between elements.

The pattern in the 4 :0 plane has been computed by this array approximation for the 18 inch Luneberg lens with 3 wavelength feed at a wavelength of 3.10 cm. The orientation of the lens and ground plane is shown in FIGS. 6 and 7. In the lens section a magnetic dipole and an electric dipole were assumed to exist in each wavelength square of surface. These elements were arranged to lie in lines parallel to the x axis. Thus the relative far-field pattern due to one line or array of magnetic dipoles is given by where m=position of a dipole in the linear array measured in integral wavelengths from the first element (m=0) in the array.

M =relative strength of magnetic dipole parallel to the y axis and at position m.

\j/ =relative phase at m with respect to some reference point (P for example).

The total pattern due to magnetic current elements is obtained by summing the E of Eq. 77 for all of the lines of magnetic dipoles necessary to cover the surface where appreciable tangential electric field exists.

Similarly the relative far-field pattern due to one line or array of electric current elements is given by 78) E (6) =cos GEJ e flu Wm) where cos 9=pattern of an electric dipole in the I =O plane. J =relative strength of electric dipole parallel to x axis at position m.

The total field due to electric current elements is obtained by summing E of Eq. 78 for all lines of electric current dipoles necessary to cover the surface where appreciable tangential magnetic field exists.

The total far field is a superposition of the fields due to the electric and magnetic dipole arrays. The relative phase and amplitude of the far field due to the planar array of electric dipoles with respect to that of the magnetic dipoles is obtained by a method outlined in the previously mentioned work entitled Endfire Slot Antennas. According to this method the far fields are adjusted so that they are in phase and their maximum values are equal. This method is exact for a lens on an infinite ground plane; it appears to be a reasonable approximation here.

Simplified Calculation of Far-Field Patterns For a line source of current of amplitude AU) and phase MI) the relative far-field pattern is given by where FQ) is the pattern of an element of source dl and the remaining symbols are defined in FIG. 16.

In general AU) and (I) may be functions such that there is no known exact solution for Eq. 79. In such cases approximate methods of evaluating Eq. 79 are required. A standard procedure would be to employ numerical integration such as the trapezoidal rule or Simp sons rule.

The approximate method used in this disclosure is that of representing a continuous source by an array of elements which have the amplitude and phase of the continuous distribution at the points where the discrete elements are located. The elements of the approximating array will have the same radiation characteristics as the differential element d! and they need not be equally spaced. Experience with dipole arrays and continuous distributions would indicate that an array with an element every half wavelength should, in most cases, give essentially the same pattern as a continuous distribution with the same amplitude and phase at corresponding points. For half wavelength spacing the integral of Eq. 79 is approximated by the series where L=length of source in wavelengths. A =amplitude at ma /2 along the source. =phase at ma /2 along the source.

Although the physical picture is that of an array of elements representing a continuous distribution, Eq. 80 is of the form that would be obtained by applying the trapezoidal rule except for the end terms of the series which are off by a factor of two.

Several computations have been made to check the accuracy of the approximation in Eq. 80 for various element spacings. The first case wnsidered is shown in FIG. 17. This is the familiar constant amplitude source for broadside radiation which, for the continuous source, has a relative far-field pattern of the form sin U E (cos r) where U =1rL cos For a broadside array of equally spaced and uniformly excited elements the relative far-field pattern is given by sin NV (82) E(COS where V=1rS cos i.

s=distance in wavelengths between elements. N =L/s.

Comparison of these patterns in FIG. 17 shows that very good agreement is obtained for spacings of a half wavelength or less and reasonably good agreement is obtained for one wavelength spacing of elements.

It is concluded that a surface-wave structure can be designed to act as a lens. In particular, a sheet of dielectric on a ground plane has been designed to operate as a Luneberg lens in the plane of the ground plane. The necessary variation in index of refraction for the Luneberg lens can be obtained through proper control of the velocity of the surface wave. Although only a TM surface-wave (vertically polarized) lens was fully discussed in this application, an analysis of the dielectric sheet on a ground plane indicates an analogous TE surface-wave (horizontally polarized) lens.

The TM surface-wave Luneberg lens with a tapereddepth antenna as a feed results in an antenna that has a lens-type pattern in the plane of the lens and an endfire pattern (modified by finite ground plane) in the orthogonal plane. In FIGS. 6, 7 and 7a, I have shown the surface-wave Luneberg lens antenna of my invention assembled in such manner so that it will protrude from the airframe, missile, or body on which it is mounted. In FIGS. 18, 19, and 20 I have shown a modified assembly of the surface-wave Luneberg lens antenna of my invention. This modified form is used when it is desired to have the surface-wave Luneberg lens antenna mounted flush with the skin or exterior surface of the airframe or missile. With this type mounting all protrusions from the surface of the airframe or missile due to the antenna mounting are eliminated and it can be appreciated that this is a very important feature in eliminating drag, aiding in maintaining uniform stability, etc., in fast moving aircraft, missiles, projectiles, etc.

In the flush mounted form of my invention the recessed lens represented at 12 is mounted flush with ground plane 13 which in turn is mounted flush with the surface of the object to which it is connected. The ground plane 13 is dished to a concave contour to maintain lens 12. A modified tapered-depth feed consisting of waveguide member 14 filled with polystyrene 15 or other like dielectric material projects through ground plane 13 and abuts the edge of lens 12 as shown at 16 with its curved apertured end which is curved to the contour of lens 12. The curved end of the tapered-depth feed which abuts the contour of the lens is shown more clearly in the enlarged views of FIGS. 21 and 22.

It is found that an approximate analysis based on optical methods can be used to obtain far-field patterns that agree reasonably well with measured patterns. The pattern in the plane of the lens is obtained from an equivalent broadside radiator that replaces the lens. The pattern in the orthogonal plane is obtained by replacing the lens and ground plane by a planar array of magnetic and electric dipoles over the antenna surface where substantial tangential E and I? exist.

The location of an effective point source feed for replacing the extended feed in the above analysis is obtained by considering the lens and feed as coupled waveguides with variable phase velocities. Neglecting mutual effects maximum coupling is assumed to exist at the point where their phase velocities are equal.

In some applications of the surface-wave Luneberg lens (e.g. reduction of the radius of rotation of the feed element) it may be desirable to feed the lens well in from the rim of the lens as shown at 17 in FIGS. 23 and 24. The exact modification in index may be obtained from the work of J. E. Eaton, entitled An Extension of the Luneberg-Type Lenses, Report 4110, Naval Research Laboratory, Washington, DC, or to a good approximation from the equation 1=\ -l- P and from Equation 59, as previously set forth.

The constants A and B are determined by the following boundary conditions:

(1) The lens is to be matched to free space.

(2) The location of the focus is to be specified.

(3) One of the emanating rays is to be made parallel to the principal ray (ray along the diameter).

By way of illustration, if the source is located at p =.851 and =18O, and the ray emanating at =45 is made parallel to the principal ray, then the constants are determined to be the following:

If the source is located at p =.65 with the other conditions remaining the same, the constants are determined to be:

The foregoing disclosure indicates that surface-wave lenses are practical and actual tests of such a lens antenna have shown it to be very practical.

While I have described my invention in certain preferred embodiments, I realize that modifications may be made and I desire that it be understood that no limitations upon my invention are intended other than may be imposed by the scope of the appended claims.

What I claim as new and desire to secure by Letters Patents of the United States is as follows:

1. A surface wave antenna comprising a relatively flat ground plane structure having an isotropic electromagnetic energy radiating surface, a radiating surface, a radiating lens having a varying index of refraction satisfying the lens relation where p is the normalized radius, c is the velocity of light in free space and v is the phase velocity of a wave in the lens of the medium, said lens further comprising a circular configuration with a flat surface in coplanar relation to and in electrical contact with said ground plane structure and a free radiating surface with radial symmetry propagation capability; and means for coupling electromagnetic energy to said lens at the focus of said circular lens.

2. A surface wave antenna as set forth in claim 1 further comprising means for controlling the velocity of said surface wave.

3. A surface wave antenna as set forth in claim 1 wherein said free radiating surface of said lens has a convex curvature relative to said ground plane and wherein the focal point of said lens is at the rim of said convex curvature.

4. A surface wave antenna as set forth in claim 1 wherein said free radiating surface of said lens has a concave curvature relative to said ground plane and wherein the focal point of said lens is at the rim of said concave curvature.

5. A surface wave antenna as set forth in claim 1 where said means for coupling electromagnetic energy to said lens couples said energy at a radial point between the center and the outer periphery of said circular lens.

6. A surface wave antenna comprising a relatively fiat ground plane structure having an isotropic electromagnetic energy radiating surface, a radiating surface, a radiating lens having a varying index of refraction satisfying the lens relation where p is the normalized radius, c is the velocity of light in free space and v is the phase velocity of wave in the lens of the medium, said lens further comprising a circular configuration with a free radiating surface having a convex curvature and radial symmetry propagation capability, and a fiat surface in coplanar relationship to and in electrical contact with said ground plane structure and the rim of said convex lens surface; and means for coupling electromagnetic energy to said lens at the rim of said lens.

7. A surface wave antenna as set forth in claim 6 wherein said means for coupling electromagnetic energy to said lens comprises a wave guide feed opening into said ground plane and abutting the base of said lens at the rim thereof.

8. A surface wave antenna comprising a relatively fiat ground plane structure having an isotropic electromagnetic energy radiating surface, a radiating surface, a radiating lens having a varying index of refraction satisfying the lens relation where p is the normalized radius, is the velocity of light in free space and v is the phase velocity of wave in the lens of the medium, said lens further comprising a circular configuration with a free radiating surface having a concave curvature and radial symmetry propagation capability, and a flat surface in coplanar relationship to and in electrical contact with said ground plane structure and the rim of said concave lens surface; and means for coupling electromagnetic energy to said lens at the rim of said lens.

9. A surface wave antenna as set forth in claim 8 wherein said means for coupling electromagnetic energy til couples said energy at the outer periphery of said concave surface of said lens.

10. A surface wave antenna comprising a relatively flat ground plane structure having an isotropic electromagnetic energy radiating surface, a radiating lens having a varying index of refraction satisfying the optical lens relation where p is the normalized radius, c is the velocity of light in free space and v is the phase velocity of a wave in the lens of the medium, said lens further comprising a circular configuration with a free radiating surface having a convex curvature and 360 radial symmetry propagation capability, and a flat surface in coplanar relationship to and in electrical contact with said ground plane structure and the rim of said convex lens surface; a feed element for coupling electromagnetic energy to said lens at the rim of said convex surface to propagate radiation as a lens in the plane of said surface structure and as an endfire antenna in the orthogonal plane.

11. A surface wave antenna comprising a relatively fiat ground plane structure having an isotropic electromagnetic energy radiating surface, a radiating lens having a varying index of refraction satisfying the optical lens relation where p is the normalized radius, c is the velocity of light in free space and v is the phase velocity of a wave in the lens of the medium, said lens further comprising a circular configuration with a free radiating surface having a concave curvature and 360 radial symmetry propagation capability, and a flat surface -in coplanar relationship to and in electrical contact with said ground plane structure and the rim of said concave lens surface; a feed element for coupling electromagnetic energy to said lens at the rim of said concave surface to propagate radiation as a lens in the plane of said surface structure and as an endfire antenna in the orthogonal plane.

References Cited in the file of this patent UNITED STATES PATENTS 2,576,182 Wilkinson Nov. 27, 1951 2,596,190 Wiley May 13, 1952 2,599,896 Clark et al June 10, 1952 2,720,588 Jones Oct. 11, 1955 2,822,542 Butterfield Feb. 4, .1958 2,869,124 Marie Jan. 13, 1959 2,875,439 Berkowitz Feb. 24, 1959 2,881,431 Hennessly Apr. 7, 1959 2,921,308 Hansen et a1. Ian. 12, 1960 2,945,230 Elliott July 12, 1960 UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3, 108,278 October 22, 1963 Carlton H. Walter In the drawing, Sheet 1, Fig. 2, the uppermost mathematical expression should appear as shown below instead of as in the patent:

in the drawing, Sheet 2, Fig. 21" should read Fig. 22 and "Fig. 22" should read Fig. 2 column 1, line 69, for "zykindrischen" read Zylindrischen line 70, for "Chemil" read chemie column 6, line 67, for "14" read 4 column 12, lines 24 to 26, for that portion of equation (56) reading "1 cos" read l-cos lines 61 and 62, after equation (62) insert a closing parenthesis; column 13, line 22, for "Eq. 51" read Eqs. 51 and 61 column 17, lines 12 to 14, equation (81), after the closing parenthesis insert column 18, line 25, the equation shoulo be numbered (83) in the left-hand margin of the column.

Signed and sealed this 11th day of August 1964.

(SEAL) Attest:

ERNEST W. SWIDER EDWARD J. BRENNER Attesting Officer Commissioner of Patents 

11. A SURFACE WAVE ANTENNA COMPRISING A RELATIVELY FLAT GROUND PLANE STRUCTURE HAVING AN ISOTROPIC ELECTROMAGNETIC ENERGY RADIATING SURFACE, A RADIATING LENS HAVING A VARYING INDEX OF REFRACTION SATISFYING THE OPTICAL LENS RELATION 